Reactant-state

In principle, the reaction cross section not only depends on the relative translational energy, but also on individual reactant and product quantum states. Its sole dependence on E in the simplified effective expression (equation (A3.4.82)) already implies unspecified averages over reactant states and sums over product states. For practical purposes it is therefore appropriate to consider simplified models for tire energy dependence of the effective reaction cross section. They often fonn the basis for the interpretation of the temperature dependence of thennal cross sections. Figure A3.4.5 illustrates several cross section models. 

Fig. 1. The rate-determining step in the neutral hydrolysis of paramethoxy-phenyl dichloroacetate. In the reactant state (a) a water molecule is in proximity of the carbonyl carbon after concerted proton transfer to a second water molecule and electron redistribution, a tetrahedral intermediate (b) is formed.

The basic chemical description of rare events can be written in terms of a set of phenomenological equations of motion for the time dependence of the populations of the reactant and product species [6-9]. Suppose that we are interested in the dynamics of a conformational rearrangement in a small peptide. The concentration of reactant states at time t is N-n(t), and the concentration of product states is N-pU). We assume that we can define the reactants and products as distinct macrostates that are separated by a transition state dividing surface. The transition state surface is typically the location of a significant energy barrier (see Fig. 1). 

For example, when the energy barrier is high compared to the thermal energy, we can assume that when a reactant state is prepared there will be many oscillations in the reactant well before the system concentrates enough energy in the reaction coordinate ... 

Given the foregoing assumptions, it is a simple matter to construct an expression for the transition state theory rate constant as the probability of (1) reaching the transition state dividing surface and (2) having a momenrnm along the reaction coordinate directed from reactant to product. Stated another way, is the equilibrium flux of reactant states across... 

That means that the transition rate is equal to the relative probability of being an activated reactant state times the average forward flux... 

What happens in a chemical reaction during the period between the initial (reactant) state and the final (product) state An answer to this question constitutes a description of the mechanism of the reaction. The study of reaction mechanisms is a major application of chemical kinetics, and most of this book is devoted to this application an introduction is given in Section 1.2. 

The chemical species in the transition state is in equilibrium with the reactant state. This assumption is discussed below. 

The rate of reaction is equal to the product of the concentration of transition state species formed from the reactant state and the frequency with which this species passes on to the product state. 

Now suppose that, from this equilibrium situation, the final state is instantaneously removed. The production of transition state species by the product state will cease. However, the production of transition state species by the reactant state is unaffected by this suppression of the final state, and, according to the third postulate of the theory, the rate of reaction is a function of the transition state concentration formed from the reactant state. This is the usual argument for the equilibrium assumption. Despite its apparent artificiality, the equilibrium assumption is generally considered to be fairly sound, with the possible exception of its application to very fast reactions. ... 

Let us now sketch the reaction coordinate diagram for the complex reaction of Scheme U, where R represents the reactant state, P the products, and I an intermediate. 

Figure 8-6. Ploi according to Fig. 8-5 of transfer free energies of the transition state (ordinate) and reactant state (abscissa) for the Menschutkin reaction of triethylamine and ethyl iodide. The reference solvent is N, Af-dimethylformamide (No. 27). Data are from Table 8-10, where the solvents are identified by number. Closed circles are polychlorinated solvents.

On the basis of the examples given above, it is reasonable to suggest that the underlying principles for optimization of the overall reaction rate with respect to the choice of metal ion are similar. That is, there are basically three states along the reaction pathway which determine the most suitable choice of metal ion. These are (1) the reactant state with bound metal and substrate before the proton transfer step, (2) the intermediately created free OH nucleophile and, (3) the subsequent transition state associated with... 

In summary one can view the ethylene epoxidation system as one where selectivity maximization requires the coexistence of the following two adsorption reactant states ... 

The rates of radical-forming thermal decomposition of four families of free radical initiators can be predicted from a sum of transition state and reactant state effects. The four families of initiators are trarw-symmetric bisalkyl diazenes,trans-phenyl, alkyl diazenes, peresters and hydrocarbons (carbon-carbon bond homolysis). Transition state effects are calculated by the HMD pi- delocalization energies of the alkyl radicals formed in the reactions. Reactant state effects are estimated from standard steric parameters. For each family of initiators, linear energy relationships have been created for calculating the rates at which members of the family decompose at given temperatures. These numerical relationships should be useful for predicting rates of decomposition for potential new initiators for the free radical polymerization of vinyl monomers under extraordinary conditions. 

The steric parameters for the estimation of reactant state effects were chosen to be the conformational free energy differences for cyclohexane axial-equatorial equilibria (A-values) (8). In order to establish the methyl group as the standard size group, modified A-values (A ) for the various groups were used, by simply subtracting the A value for the methyl group (1.70) from the A values of the various substituents ... 

Equation 6 would hold for a family of free radical initiators of similiar structure (for example, the frarw-symmetric bisalkyl diazenes) reacting at the same rate (at a half-life of one hour, for example) at different temperatures T. Slope M would measure the sensitivity for that particular family of reactants to changes in the pi-delocalization energies of the radicals being formed (transition state effect) at the particular constant rate of decomposition. Slope N would measure the sensitivity of that family to changes in the steric environment around the central carbon atom (reactant state effect) at the same constant rate of decomposition. 

Temperature error differences (AT), equal to the experimental temperature minus the linear regression temperatures, were then plotted by another linear regression analysis against the EA values to obtain the reactant state effect slope ... 

For the transition state effect correlation equation (equation 7), the slopes (M) for the various reactions of this study are positive in sign (Tables II and V). When the negative AE(x) values (Table I) are multiplied by the slope, for a particular initiator at a given rate, a negative (reaction temperature-lowering) value is obtained. For the reactant state effect correlation equation (equation 8), the slopes (N) for the various reactions of this study are negative in... 

The best fits to the linear equation 8, for temperature differentials (from equation 7) versus reactant state steric effects, are obtained for reaction 4 (Table III). A modest correlation for equation 8 is obtained for reaction 1. Essentially no fit to equation 8 is found for reactions 2 and 3 (small correlation coefficients and small N slopes). 

It is not intended that the equations of this study be used to supplant the much more elegant molecular orbital calculations, both semiempirical and ab initio, and the mechanical modeling studies of radical forming reactions. However, it may be possible to make some hypotheses about differences in mechanisms between reaction families, based on the values of the slopes in Table IV. The slopes could be considered "sensitivity factors" (like rho values) for measuring the relative magnitude of transition state effects (U) and reactant state effects (N) on the rates of the four reactions of this study. 

From Table IV the relative magnitudes of the reactant state "sensitivity factor" (N) are 4>1>2=3= zero. From this analysis the decomposition rates of traiw-phenyl, alkyl diazenes (2) and iert-butyl peresters (3) can be predicted by assuming a dependence only on transition state effects, with no need to incorporate the back strain of the reactants into the equation. 

Irons-phenyl, alkyl diazenes (2), peresters (3) and hydrocarbons (4). These equations are intended to be used for their predictive value for applications especially in the area of free radical polymerization chemistry. They are not intended for imparting deep understanding of the mechanisms of radical forming reactions or the properties of the free radical "products". Some interesting hypotheses can be made about the contributions of transition state versus reactant state effects for the structure activity relationships of the reactions of this study, as long as the mechanisms are assumed to be constant throughout each family of free radical initiator. 

One notes that the proportionality constant, a, depends on the reaction energy, AEy. Therefore, Eq. (1.3) is not strictly a linear relation between activation energy change and reaction energy. In the extreme limit of high exothermicity of the reaction energy a = 0, and the crossing point of the two curves is at the minimum of curve Vj. In this case the transition state is called early. Its structure is close to that of the reactant state. 

The results here clearly demonstrate some of the important differences between reactions in the vapor phase and those in the aqueous phase. Water solvates the ions that form and thus enhances the heterolytic bond activation processes. This leads to more significant stabilization of the charged transition and product states over the neutral reactant state. The changes that result in the overall energies and the activation barriers of particular elementary steps can also act to alter the reaction selectivity and change the mechanism. 

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